resonant dynamics of gravitationally bound pair of binaries...

MNRAS 475, 5215–5230 (2018) doi:10.1093/mnras/sty132Advance Access publication 2018 January 15

Resonant dynamics of gravitationally bound pair of binaries: the case of1:1 resonance

Slawomir Breiter1‹ and David Vokrouhlicky2

1Astronomical Observatory Institute, Faculty of Physics, Adam Mickiewicz University, Sloneczna 36, PL-61-286 Poznan, Poland2Institute of Astronomy, Charles University, Prague, V Holesovickach 2, CZ-180 00 Prague 8, Czech Republic

Accepted 2018 January 10. Received 2018 January 8; in original form 2017 December 8

ABSTRACTThe work presents a study of the 1:1 resonance case in a hierarchical quadruple stellar systemof the 2+2 type. The resonance appears if orbital periods of both binaries are approximatelyequal. It is assumed that both periods are significantly shorter than the period of principal orbitof one binary with respect to the other. In these circumstances, the problem can be treated asthree independent Kepler problems perturbed by mutual gravitational interactions. By meansof canonical perturbation methods, the planar problem is reduced to a secular system with1 degree of freedom involving a resonance angle (the difference of mean longitudes of thebinaries) and its conjugate momentum (involving the ratio of orbital period in one binaryto the period of principal orbit). The resonant model is supplemented with short periodicperturbations expressions, and verified by the comparison with numerical integration of theoriginal equations of motion. Estimates of the binaries periods variations indicate that theeffect is rather weak, but possibly detectible if it occurs in a moderately compact system.However, the analysis of resonance capture scenarios implies that the 1:1 resonance should beexceptional amongst the 2+2 quadruples.

Key words: methods: analytical – celestial mechanics – binaries: eclipsing – stars: kinematicsand dynamics.

1 IN T RO D U C T I O N

The exact statistical census of stellar hierarchies, from binaries tomultiple-star systems, is subject to debate because of difficulties inprecise debiasing procedure of surveys. Nevertheless, a significantprogress has been achieved over the past decade or so. We know thatstars like companions, and nearly 10 per cent live in hierarchies ofmore than two (e.g. Tokovinin 2014; Riddle et al. 2015). This num-ber was determined for solar-type stars of F- and G-classes, and it islikely quite larger for more massive stars. As outlined in Tokovinin(2008), relative number of systems with different multiplicity andtheir parameters (such as ratio of masses of participating stars andorbital periods of their sub-systems) put important constraints ontheir formation processes. These include not only direct fragmenta-tion of the initial cloud but also orbital processes such as gas- andtide-driven migration. The latter leave important fingerprints on thepresently observed architecture of the stellar systems.

In this paper, we focus on the quadruple systems. Their long-termstability requires one of the two architectures: (i) ((2 + 1) + 1),namely a triple system accompanied with a distant star, or (ii)(2 + 2), namely two binary systems revolving about a common

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centre of mass. Of these two, we are concerned about the lattercategory (ii). Interestingly, the 2+2 systems are more frequent (e.g.Tokovinin 2008; Riddle et al. 2015) and have distinct properties,both factors suggesting their unique formation route. In particular,Tokovinin (2008) found that (i) the binaries embodied in the 2+2quadruples are frequently composed of stars with similar mass, (ii)orbital spin of binaries and their common orbit in compact 2+2quadruples are correlated (unlike in triples and wider 2+2 quadru-ples), and (iii) period ratio of the binaries in compact 2+2 quadru-ples is within an order-of-magnitude near unity (unlike in triplesand wider 2+2 quadruples).

Unlike the primordial dynamical issues mentioned above, wedeal with the present dynamical configuration. Equations of seculardynamics of quadruple systems in the 2+2 hierarchy have been for-mulated and studied at different levels of accuracy and with differ-ent goals (e.g. Pejcha et al. 2013; Hamers & Portegies Zwart 2016;Vokrouhlicky 2016; Hamers & Lai 2017). Put in an immediate prac-tical use, these results may be employed to interpret observationsof long-term changes in architecture of these quadruple systems,such as precession of periapses of the two participating binaries orchanges in orientation of their orbital planes witnessed by variationsof eclipse depths (e.g. Pribulla et al. 2008; Zasche & Uhlar 2013,2016). This is actually a traditional framework used for decades forinterpretation of orbital changes in triple stellar systems, only now

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5216 S. Breiter and D. Vokrouhlicky

adapted for a more general architecture.1 However, as the availableobservational data about the 2+2 quadruple systems grow, prob-lems of new flavor may appear. This is the case of a possibility ofmean motion resonances between the two participating binary sys-tems. Obviously, the previously developed secular approach cannotbe used to tackle this problem and new tools are needed.

On the observational side, the case for such resonances has notyet been established exactly. This is because it would require acomplete information about the architecture of the system. We misssuch a detailed knowledge for most of the triple systems, so it is notsurprising that the data about 2+2 quadruples are equally incom-plete. Yet, strong hints and suggestions were recently published.For instance, Ofir (2008) suggested that the system BI 108 in theLMC field of OGLE survey is likely a doubly-eclipsing system in2+2 architecture with orbital periods PA � 3.577 98 d and PB �5.366 55 d for the two binaries. Ofir (2008) also noted that the pe-riods are within �0.03 per cent near the 3:2 ratio and conjecturedthat the system is in the corresponding dynamical resonant state.This commensurability of the two periods in the BI 108 systemhas been confirmed, and even strengthened, by Kolaczkowski et al.(2013). While discussing also other possibilities for occurrence ofthe two periods (see also Rivinius, Mennickent & Kołaczkowski2011), these authors concluded that the resonant state of thetwo interacting binaries is the most likely conclusion. Similarly,Cagas & Pejcha (2012) studied a double period, doubly-eclipsingsystem CzeV343 with PA � 0.806 931 d and PB � 1.209 373 d.Again, the ratio PB/PA is within �0.1 per cent equal to 3:2, sugges-tive for a dynamically resonant configuration. We also note that inboth cases the longer-period orbit has been found slightly eccentric,� 0.08 and � 0.18. Given the first-order of the 3:2 resonance, thisis a necessary pre-requisite of the physical resonance between thetwo binaries. On the other hand, the two systems are unresolvedand, worse, the long period of the putative revolution of the centreof mass was not detected yet. It is not surprising then, that otherinformation, such as the mutual inclination of the binary orbitalplanes, about the two systems is also not available.

Apart from these two strongest candidates for 3:2 resonant sys-tems, we note that the analysis of the eclipsing binaries in theOGLE-III SMC fields by Pawlak et al. (2013) revealed 15 candi-dates for 2+2 quadruples. Two of them have the period ratio closeto 3:2 within 1–2 per cent. Finally, we note a rare case of a doublyeclipsing unresolved Kepler system KIC 4247791. The photometricand spectroscopic analysis of Lehmann et al. (2012) revealed that itconsists of two, possibly coplanar binaries on nearly circular orbitswith periods PA � 4.04973 d and PB � 4.10087 d. This is withinabout a percent level close to their 1:1 ratio. However, as before,the data of this interesting system do not presently allow to revealthe mutual motion of the two binaries about their common centreof mass.

With the possible caveat that the periods in the previously men-tioned systems are near-commensurate by pure chance, we take themove on the theory side and investigate conditions for mean motionsresonances in the 2+2 quadruple systems. It is interesting to notethat such a resonant configuration has not been studied in the past.

1 It is interesting to note that Brown (1937), in his series of pioneeringworks, analysed these long-period variations for the system ξ UMa, whichis actually a quadruple in the 2+2 architecture. However, one of the binaryperiods is much shorter than the other, so the short-period system may beconveniently replaced with a single body from the dynamical point of viewand the system effectively treated as triple with sufficient accuracy.

This is for two reasons. First, as described above, the observationalhints about such resonances appeared in the literature only veryrecently. Secondly, it has been actually customary that problemsof stellar orbital dynamics were originally built on analogues fromour planetary system. For instance, early studies of perturbations intriple systems were directly developed from tools of Delaunay andHill-Brown theories of lunar motion (see Brown 1936a,b,c, 1937).In our case, however, the problem is autonomous to stellar systemsand no planetary analogy exists. In this sense, while using classicaltools of perturbation theory, our approach is a novel contribution toorbital mechanics.

As discussed above, the current observations motivate to studymainly the 3:2 mean motion resonance between the orbital motionsof the two binaries. To set the stage and gain first insights with theproblem, we however start with simpler, but also stronger, 1:1 res-onance. This case requires the least analytic labour because, unlikethe first-order resonances, like 3:2, it exists for circular, or near-circular, orbits of the two binaries. Unless there were reasons forthe eccentricities to be triggered to large values by other dynamicaleffects, the resonant problem can be very well approximated with aone degree of freedom, time-independent Hamiltonian model (Sec-tions 2 and 3). Even if the solution of the reduced model remainsin formal quadratures, not being expressible in terms of elementaryfunctions, the principal characteristics such as resonant equilibriaor resonant width can be easily determined (Section 4). The modelhas been validated by the comparison with numerical integrationof original equations of motion (Section 5), which required the ad-dition of periodic perturbations formulae (Appendix A) and someanalytical estimates of the secular evolution of eccentricities (Ap-pendix B).

One of the most interesting aspects of the putative mean motionresonances in the 2+2 stellar quadruples is that they very likelyoriginate from a capture (see Cagas & Pejcha 2012). This is becausethe volume occupied by the resonances in the orbital phase spaceis extremely small. Thus, the probability that the systems happenedto be born in a resonant state is very low. Rather, the resonanceshould have been established during the tidal evolution of PA or PB.Therefore, the statistics of the occurrence of resonant cases could,in an ideal situation when all other biases are removed, be linkedto parameters of the tidal evolution of the binary orbits. For thisreason, we also pay attention to the resonant capture conditions andprobability (Section 6).

2 DY NA M I C A L F R A M E WO R K

2.1 Variables and Hamiltonian

Consider a quadruple system made of two binary subsystems Aand B (Fig. 1 ). The binary A includes two stars having masses m1

and m2, with a relative position vector rA directed from m1 to m2.Similarly, the relative position vector rB points from the star of themass m3, to the second component of the binary B, having the massm4. Then, the position vector R locates the centre of mass of thesubsystem B with respect to the centre of mass of A. Since all threevectors are relative, they can be reckoned in any reference framewith fixed directions of axes.2 By assumption, we attach the symbols

2 We have decided to abbreviate the notation used by Vokrouhlicky (2016),replacing his subscripts Aa, Ab, Ba, Bb with 1, 2, 3, and 4, respectively, andomitting the subscript AB.

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Resonant dynamics of pair of binaries 5217

Figure 1. Geometry of the planar quadruple system. True longitudes areFA = fA + � A and FB = fB + � B.

so that, whenever the masses are different, m1 > m2, m3 > m4, andMA > MB, where

MA = m1 + m2, MB = m3 + m4, (1)

are the total masses of binaries A and B. With this convention, thesystem can be parametrized by three mass ratios

μA = m2

MA, μB = m4

MB, μ = MB

M, (2)

where M = MA + MB, is the total mass of the system, with each of

the coefficients belonging to the interval (0, 1/2].Attaching to rA, rB, and R momentum vectors pA, pB, and

P , we obtain a canonical set sometimes called ‘hierarchical Ja-cobi coordinates’ (e.g. Milani & Nobili 1983; Beust 2003). Indeed,the variables offer the advantages of the standard Jacobi variables,widespread in hierarchical triples studies, being well adapted to thehierarchy in question, having the momenta tangent to trajectories,and achieving the reduction from N to N − 1 body problem in thecanonical framework.

The Hamiltonian function of the problem, as given byVokrouhlicky (2016), may be decomposed into three distinct parts:

H = HK

(rA, rB, R, pA, pB, P

) + Hint (rA, rB, R)

+Hc (rA, rB, R) . (3)

The principal term

HK = Hb + Hp, (4)

where

Hb =[

p2A

2 m′A

− Gm′

AMA

rA

]+

[p2

B

2 m′B

− Gm′

BMB

rB

], (5)

Hp = P2

2 m′ − Gm′MR

, (6)

with gravitation constant G, and reduced mass parameters

m′A = m1m2

MA= μA(1 − μA)(1 − μ)M,

m′B = m3m4

MB= μB(1 − μB)μM,

m′ = MAMB

M= μ(1 − μ)M, (7)

defines three decoupled two-body problems: the motion of m2 withrespect to m1 in the binary A, the motion of m4 around m3 in thebinary B (both the binaries in Hb), and the motion of the barycentreof B with respect to the barycentre of A (inHp) – each one followinga Keplerian conic in the absence of other terms in H.

The union of three Keplerian motions is perturbed by the interac-tions potential. There one may distinguish two parts: Hint and Hc.The term

Hint = −Gm′MR

∑n≥2

χA,n

( rA

R

)n

Pn (γA)

− Gm′MR

∑n≥2

χB,n

( rB

R

)n

Pn (γB) , (8)

may be interpreted as defining the perturbations in a fictitious tripleincluding the barycentre of B and the ‘dipole’ m1, m2 (first line), orin a fictitious triple system including the barycentre of A and the‘dipole’ m3, m4 (second line). There, the Legendre polynomials Pn

have arguments

γA = cos SA = R · rA

R rA, γB = cos SB = R · rB

R rB, (9)

and the mass functions χ are

χA,n = μA (1 − μA)[μn−1

A − (μA − 1)n−1],

χB,n = μB (1 − μB)[μn−1

B − (μB − 1)n−1]. (10)

Adding either line of equation (8) to HK, one might obtain atypical, hierarchical three-body problem (e.g. Harrington 1969;Breiter & Vokrouhlicky 2015), yet the coupling of both lines throughR should not be forgotten in a non-linear approximation. But thedirect coupling between all members of A and B is contained in theremainder Hc, whose leading term reads (Hamers et al. 2015)

Hc = −3 Gm′Am′

B

4 R

( rArB

R2

)2

× (1 − 5 γ 2

A − 5 γ 2B + 2 γ 2 + 35 γ 2

A γ 2B − 20γ γAγB

), (11)

with

γ = cos S = rA · rB

rA rB. (12)

The readers should be warned that the analogical equation (10) inVokrouhlicky (2016) contains an error, which makes it valid onlyin the planar case.

The equations of motion resulting from the Hamiltonian (3) arenon-integrable, but if the binaries A and B are sufficiently distantfrom each other, i.e. rA + rB � R, we can resort to perturbationtechniques, considering perturbed Kepler problems.

2.2 Preliminary simplifications

Experience suggests that mean motion related resonances arestrongest when the orbits are coplanar.3 Thus, we restrict the

3 Moreover, observational data about the quadruples in the 2+2 hierarchyindicate near-coplanarity of many systems (e.g. Tokovinin 2008).



discussion to the planar case, where all three angular momentumvectors rA × pA, rB × pB, and R × P have the same direction (andtheir scalar products are all positive).

In the plane perpendicular to the angular momenta, the thirdcomponents of the position and momentum vectors are null, and thesystem has 6 degrees of freedom. The angles occurring in equations(9) and (12) become related in an elementary manner:

S = SA − SB, (13)

which allows to formulate equation (11) in terms of only two of thethree angles.

Another simplification amounts to dropping from Hint the termswith n > 3. This requires an explanation, because of an apparentinconsistence with the order of magnitude ofHc from equation (11).By analogy with Breiter & Vokrouhlicky (2015), let us introducea small parameter ε defined by the square root of ratio of Hp toHb, i.e. ε = √

max (rA, rB)/R. Then we have Hb = O(1), Hp =O(ε2), Hint(n = 2) = O(ε6), Hint(n = 3) = O(ε8), and Hint(n =4) = O(ε10). If the normalizing canonical transformation is sought,then non-linear terms in the Hamiltonian appear at O(ε9), as inBreiter & Vokrouhlicky (2015, Section 3.1). Having decided toconsider only linear perturbations in the present work, we will missthe terms of order 9, hence there is no point in considering n = 4terms inside Hint, which are of order 10.

This argument, however, does not apply to Hc – more precisely,to its resonant term amplitude. The resonance introduces a denom-inator of the order of the square root of the amplitude (Garfinkel1970), which means that the effect might be O(ε5) – comparablewith the influence of Hint(n = 2).

Finally, we assume that the eccentricities of unperturbed orbitsin the subsystems A and B are small, although the ‘principal orbit’,as we are going to call the one related with R, P remains elliptic.The small eccentricity assumption means that we are going to retainonly the first powers of osculating eA and eB in the Hamiltonian,and set the mean eccentricities eA = eB = 0 after the averagingtransformation. Let us emphasize that this particular approximationis allowed only in the 1:1 resonance case, since all other meanmotion resonances, for example 3:2, require non-zero eccentricityto appear.

Having restricted the motion to the planar and almost circularcase, we can introduce a subset of the modified Delaunay variables.Selecting an arbitrary departure direction in the plane of motion,we first position the pericentre of the principal orbit using the lon-gitude of pericentre � (see Fig. 1), and define the position of thebarycentre of B on the orbit through the mean anomaly l. This leadsto the canonically conjugate coordinate-momentum set (λ, q, , Q),where

λ = l + �, = m′√GMa = m′na2,

q = −�, Q = (

1 −√

1 − e2)

. (14)

The mean motion symbol n has a purely formal meaning, definedthrough n2a3 = GM, as a function of the semi-axis a; actual rateof l differs from n because of the perturbations, but we will use theterm ‘unperturbed principal period’ for the quantity

P = 2π

n. (15)

For the binaries A and B, we introduce

λA = lA + �A, A = m′A

√GMAaA = m′

AnAa2A,

qA = −�A, QA = A

(1 −

√1 − e2

A

), (16)

and

λB = lB + �B, B = m′B

√GMBaB = m′

BnBa2B,

qB = −�B, QB = B

(1 −

√1 − e2

B

), (17)

where the osculating mean longitudes λA, λB and pericentres lon-gitudes � A, � B are measured from the same departure directionas � . The mean motions nA and nB are linked with the semi-axesaA, aB through n2

Aa3A = GMA, n2

Ba3B = GMB; they may serve to

introduce the unperturbed periods

PA = 2π

nA, PB = 2π

nB. (18)

If f stands for the true anomaly of the principal orbit (an implicitfunction of all canonical variables λ, q, , Q), the mutual positionangles are

SA = fA − f + �A − �,

SB = fB − f + �B − �,

S = fA − fB + �A − �B, (19)

where the true anomalies are functions of eccentricities and meananomalies

fA = lA + 2eA sin lA + O(e2A), fB = lB + 2eB sin lB + O(e2

B).

(20)

Expressing the Hamiltonian in these variables, we find the Kep-lerian part HK as a sum of

Hb = − (GMA)2(m′

A

)3

22A

− (GMB)2(m′

B

)3

22B

, (21)

Hp = − (GM)2(m′)3

22. (22)

In the truncated perturbing potential

Hint = −Gm′MR

3∑n=2

χA,n

( rA

R

)n

Pn (cos SA)

− Gm′MR

3∑n=2

χB,n

( rB

R

)n

Pn (cos SB) , (23)

we retain the radius R = a(1 − e2)(1 + ecos f )−1 as an exact implicitfunction of the canonical variables (14), whereas for the remainingtwo radii

rA = aA (1 − eA cos lA) + O(e2A),

rB = aB (1 − eB cos lB) + O(e2B). (24)

The remaining part Hc can be transformed into


B

32 R

( rArB

R2

)2(6 + 10 cos 2SA + 10 cos 2SB

+ 35 cos 2(SA + SB) + 3 cos 2(SA − SB)) . (25)

The last term in (25) has an argument with possibly vanishingfrequency, so we modify the set (16,17) introducing canonical res-onance variables

ψ1 = λA − λB, �1 = A, (26)

ψ2 = λB, �2 = A + B. (27)

According to the small eccentricities postulate, for the mean vari-ables ψ1 ≈ SA − SB, and ψ2 ≈ SB + f + � , with both equalitiesexact if eA = eB = 0.



3 LI N E A R SE C U L A R M O D E L

3.1 First averaging

In the first step, we average the Hamiltonian with respect to the angleψ2 – the one with the presumably shortest unperturbed period. Inother words, we perform a transformation of variables resulting in apartial normalization of the Hamiltonian with respect to the leadingterm Hb, which now reads

Hb = − (GMA)2(m′

A

)3

2�21

− (GMB)2(m′

B

)3

2 (�2 − �1)2 . (28)

Being more specific on the meaning of ‘partial’, we construct thetransformation so that the new Hamiltonian belongs to the kernelof the Lie derivative

L2 = ∂Hb

∂�2

∂

∂ψ2= nB

∂

∂ψ2, (29)

instead of the complete

Lb = ∂Hb

∂�1

∂

∂ψ1+ ∂Hb

∂�2

∂

∂ψ2= (nA − nB)

∂

∂ψ1+ nB

∂

∂ψ2. (30)

Remaining in the linear approximation, we obtain the Hamilto-nian in new variables through a simple averaging with respect toψ2. Actually, the new variables should have different symbols, butwe retain the old ones and distinguish the initial, osculating setfrom the new ‘intermediate variables’ by the context. Thus, afterthe averaging we obtain HK(�1, �2, ) = HK(�1, �2, ), and,

Hint = −GMm′

4R

χA,2a2A + χB,2a

2B

R2, (31)


B a2Aa2

B

32R5cos 2ψ1, (32)

so that L2H = 0. Note that in both Hint, and Hc, we have seteA = eB = 0. Additionally, the ψ1 independent part of Hc has beendiscarded. As a result, the intermediate �2 = A + B becomesa new constant of motion (although both A, B, hence the semi-axes aA, aB, remain variable). The transformation is defined by thegenerating function S1, whose leading part, derived from Hint(n =2) alone, is

S1 ≈ S1,A + S1,B, (33)

where

S1,ν = GMm′

8R3

χν,2a2ν

nν

[3 sin 2(F−λν)−18eν sin (2F−λν−�ν)

+ 4eν sin (λν−�ν) + 2eν sin (2F − 3λν + �ν)] , (34)

and ν stands for A or B. For brevity, we have retained the symbols λν ,aν as implicit functions of ψ1, ψ2, �1, and �2. The true longitudeon the principal orbit is

F = f + �. (35)

3.2 Second averaging

The second transformation will normalize the Hamiltonian H withrespect to Hp, so that the new function H is independent on λ, i.e.belongs to the kernel of

Lp = ∂Hp

∂

∂

∂λ= n

∂

∂λ. (36)

Again, we remain within a linear perturbation theory domain, whichmeans that the Hamiltonian in new variables is obtained through anelementary averaging with respect to λ, actually performed throughthe substitution rule

1

2π

∫ 2π

0Y dλ = 1

2π

∫ 2π

0

YR2

a2√

1 − e2df , (37)

for any function Y explicitly depending on the true anomaly f.Thus, we find, in the new variables (named ‘mean variables’), theHamiltonian

HK = HK(�1, �2,), (38)

Hint = −GMm′

4a3η3

(χA,2a

2A + χB,2a

2B

), (39)


B a2Aa2

B(2 + 3e2)

64a5η7cos 2ψ1, (40)

where

η =√

1 − e2. (41)

In the mean variables, the system has one degree of freedom, ad-mitting four integrals of motion: = const, Q = const, �2 = const,

and H(ψ1, �1, �2, ) = const. In terms of mean Keplerian ele-ments of the principal orbit, it implies constant a and e. The gener-ator S2 of the second transformation is again the sum of two similarparts

S2 ≈ S2,A + S2,B, (42)

where

S2,ν = −GMm′χν,2a2ν

4na3η3(f − l + e sin f ) . (43)

3.3 The resonant Hamiltonian

A final fine tuning of the Hamiltonian aims at obtaining the simplest

possible form. Since H actually involves two constant action vari-ables and �2, we can use their values as free parameters defininginitial conditions. In particular, we are going to use as a scalingparameter: dividing the momenta and the Hamiltonian by the sameconstant (let it be ) conserves the structure of canonical equationsof motion, even if the transformation is not strictly symplectic (itis not univalent). Under this scaling, we introduce dimensionlessmomenta J = �1/, W = �2/, leaving the old angles under thenew names, so that the variables pair to be further studied is

ϕ = ψ1, J=�1

=m′

A

m′

√MAaA

Ma=m′

A

m′

(MA

M

) 23(

PA

P

) 13

, (44)

and, by the definition, we restrict the values of J to the range

0 < J � W = J + m′B

m′

√MBaB

Ma

= J + m′B

m′

(MB

M

) 23(

PB

P

) 13

. (45)

This scaling will be followed by the introduction of a new indepen-dent variable τ , related to time t through

nt = τ, n = (GM)2(m′)3

3, (46)



so that �τ = 2π is equivalent to �t = P.

Summing up, the resonant Hamiltonian K is obtained from H by

dropping the �1-independent part Hb, dividing the rest by factorn, and substituting �1 = J, �2 = W, and ψ1 = ϕ. The resultis

K = K0 + K2 + K′4, (47)

K0 = − A01

2J 2− A02

2(W − J )2, (48)

K2 = A21J4 + A22(W − J )4, (49)

K′4 = A4J

4(W − J )4 cos 2ϕ, (50)

where

A01 = (1 − μ)2(1 − μA)3μ3A

μ3= A,

A02 = μ2(1 − μB)3μ3B

(1 − μ)3= B,

A21 = − μ4

4η3(1 − μ)2(1 − μA)3μ3A

= − μ

4η3A,

A22 = − (1 − μ)4

4η3μ2(1 − μB)3μ3B

= −1 − μ

4η3B,

A4 = − 9(2 + 3e2)(1 − μ)2μ2

64η7(1 − μA)3μ3A(1 − μB)3μ3

B

= −9(2 + 3e2)(1 − μ)μ

64η7AB. (51)

Introducing a constant C, defined by

C = μB(1 − μB)

μA(1 − μA)

(μ

1 − μ

) 53

, (52)

we may observe that

W

J= 1 + C

(PB

PA

) 13

, (53)

andB

A= C3. (54)

The complete equations of motion resulting from the resonantHamiltonian K are

dϕ

dτ= ∂K

∂J= A01

J 3+ A02

(J − W )3

+ 4A21J3 − 4A22(W − J )3

+ 4A4J3(W − J )3(W − 2J ) cos 2ϕ, (55)

dJ

dτ= −∂K

∂ϕ= 2A4J

4(W − J )4 sin 2ϕ. (56)

In the next section, we analyse the properties of the resonant Hamil-tonian (47) and the solution of equations (55) and (56) for the ‘res-onant variables’ ϕ, J.

4 R E S O NA N C E P RO P E RT I E S

4.1 Equilibria

The system defined by K behaves in a pendulum-like manner, asseen in Fig. 4. Regardless of the mass ratios, it admits four equi-libria resulting from the condition dϕ

dτ= dJ

dτ= 0. Two of them, with

ϕ = 0, or ϕ = π, and a common value Ju are unstable. Other two,with ϕ = π/2, or ϕ = 3π/2, and the momentum Js, are stable.Thus, the motion with rA permanently perpendicular to rB (up tosome presumably small periodic perturbations) is stable, whereaspermanently parallel rA and rB are not.

Although in general Js = Ju, a kind of degeneracy occurs whenboth binaries have the same total mass and the mass ratios in bothbinaries are equal, i.e. μA = μB, and μ = 1

2 ; we will refer to suchsystems as ‘symmetric’. The special case of four equal masses,with μA = μB = μ = 1

2 , will be called ‘completely symmetric’.Symmetric systems have the ratio C = 1, hence A = B in thecoefficients of the Hamiltonian K. According to equations (51), itmeans that A01 = A02 and A21 = A22, leading to symmetry in thecoefficients of J and (W − J).

In symmetric systems, the right-hand side of equation (55) be-comes null at the exact value

J∗ = Js = Ju = W

2, (57)

thanks to the presence of the factor (W − 2J), independent of ϕ.According to equation (53), with C = 1, it means that both types ofthe equilibria are exactly defined by the ratio of unperturbed periodsPB/PA = 1.

For arbitrary masses, however, the situation is less favourable:we have to face an algebraic equation of degree 13, which impliesthe use of approximate methods. The first approximation to theresonant values of the momentum is Js ≈ Ju ≈ J∗, derived as theroot of

∂K0

∂J= 0, (58)

and easily found as

J∗ = W

1 + C. (59)

Given the W, this value of J∗ refers simply to the equal unper-turbed periods PA = PB, as follows from equation (53). For furtherconvenience, we can add that

W − J∗ = CW

1 + C. (60)

In order to improve this estimate, we have applied a stan-dard perturbation technique assuming that K0 = O(1), K2 = O(ε),K2 = O(ε2), and substituting

J = J∗(1 + εJ1 + ε2J2

). (61)

The parameter ε serves only for the ordering purpose and ultimatelyit is set to 1. The correction J1, common for both the equilibria, isof the order of W6

J1 = 4C(A21 − A22C

3)W 6

3A3(1 + C)7= C(1 − 2μ)W 6

3(1 + C)7A2η3. (62)

The second correction is the sum

J2 = Jc ± Jp, (63)

where the first term, similar for both equilibria,

Jc = C(1 − 2μ)(−5 + 2C + 7(1 − C) μ)W 12

9A4(1 + C)14η6, (64)

is typically negligible. The difference between the stable and unsta-ble equilibrium locations is given by

Jp = 3(1 − C)C(2 + 3e2

)(1 − μ)μW 10

16A3(1 + C)11η7, (65)



so that

Ju ≈ J∗(1 + J1 + Jc + Jp

), Js≈J∗

(1 + J1 + Jc − Jp

), (66)

and Ju − Js = 2Jp ≥ 0.As expected, all corrections vanish for symmetric systems, where

either (1 − 2μ), or (1 − C) become null. Readers should be warned,that the above approximations fail if the system is too compact(small W assumption violated) or if we leave the domain of stellar-type systems (e.g. abnormally large values of C when μA is toosmall)

4.2 Libration amplitude and period

4.2.1 Oscillator and pendulum approximations

Elementary estimates of the resonance strength and time-scale canbe obtained using a simple pendulum model. In order to constructit, we first introduce a shifted momentum D = J − Js ≈ J − J∗, andexpand K in Taylor series of D, dropping all powers higher than 2.Observing the orders of magnitude of subsequent terms, we retainonly

K∗(ϕ,D) = −3

2

(A01

J 4∗+ A02

(W − J∗)4

)D2

+ A4J4∗ (W − J∗)4 cos 2ϕ. (67)

The half width of the libration zone � (an upper bound on theamplitude of libration in J) is found by equating the Hamiltonianvalues with ϕ = 0 and ϕ = π/2, i.e. K∗(0, 0) = K∗(π/2, �). Theresult is

� = 2J 4∗ (W − J∗)4

√−A4

3(A02J 4∗ + A01(W − J∗)4

)

≈ C W 6

4A (1 + C)6η3

√3

(2 + 3e2

)(1 − μ) μ

A (1 + C) η, (68)

where the second line results from the substitution of equation(59). Thus, the maximum amplitude of the libration in J is directlyproportional to the sixth power of W (hence to the squares of theperiod ratios PA/P and PB/P).

Relating � to more easily observable quantities, we can use thedefinitions (44) and (45) to establish the rule

δPA

PA= 3

�

J∗= −C

δPB

PB, (69)

linking the amplitude of J with a relative change of period of thebinary A or B.4

Then, the maximum relative variation of the nominal period inthe binary A is

δPA

PA= 3A

16

4

√3

(2 + 3e2

)(1 − μ) μ

(1 + C) η7

(PA

P

) 53

. (70)

For a completely symmetric system with a circular principal orbitit simplifies to

δPA

PA≈ 0.36

(PA

P

) 53

. (71)

4 Strictly speaking, δPA/PA = −C(PB/PA)13 δPB/PB, but in the vicinity of

resonance we assume that the reference periods are equal.

Figure 2. Maximum relative variation of period for a completely symmetric(all masses equal) system as a function of period ratio PA/P.

The minimum libration period (or the upper bound of the librationfrequency) can be derived from the harmonic oscillator approxima-tion in the vicinity of a stable equilibrium. To this end, we introducea new angle d = ϕ − π/2 and substitute cos 2ϕ ≈ −1 + 2d2 intoequation (67). After dropping a constant term, we obtain the Hamil-tonian

Kh(d, D) = −3

2

(A01

J 4∗+ A02

(W − J∗)4

)D2

+ 2A4J4∗ (W − J∗)4d2, (72)

generating the harmonic oscillations of d and D with a frequency

ω∗ =√

−12A4

(A01(W − J∗)4 + A02J 4∗

)

≈ 3

4

(W

1 + C

)2√

3(2 + 3e2

)(1 − μ) μ

Aη7. (73)

Recalling the time unit choice made in equation (46), we notice thatthe dimensionless ω∗ actually gives the ratio of the principal orbitperiod P to the small amplitude libration period P∗, so

P∗ = P

ω∗, (74)

regardless of the time units used in P. Thus, in particular, we havethe minimum libration period

P∗P

≈ 1.37

(P

PA

) 23

, (75)

for a completely symmetric system with e = 0.

4.2.2 Complete secular model

Let us confront the estimates provided in Section 4.2.1 with thenumerical analysis of the complete secular equations of motion(equations 55 and 56) derived from the Hamiltonian equation (47).

In order to find the maximum relative variations of orbital periodsdue to the resonance, we numerically solve a system of equations⎧⎨⎩

KJ (0, Ju) = 0,

KJ (π/2, Js) = 0,

K(0, Ju) = K(π/2, Js + �),(76)

where KJ stands as an abbreviation for the partial derivative of Kwith respect to the momentum J. Having fixed the mass ratios andthe momentum W, we find a unique quadruple (Js, Ju, �−, �+),



Figure 3. Maximum relative variations of periods for m1 = 1.70 M�,m2 = 1.54 M�, m3 = 1.46 M�, and m4 = 1.35 M�, as a function ofperiod ratios PA/P and PB/P.

where �± are the two real roots closest to the stable equilibriummomentum Js: negative �− and positive �+. These quantities canbe converted into nominal periods by means of relations

PA(J ) =(

μJ

(1 − μA) μA

)3P

(1 − μ)2, (77)

PB(J , W ) =(

(1 − μ) (W − J )

(1 − μB) μB

)3P

μ2, (78)

resulting from the definitions (44) and (45). Thus, with a set ofsolutions on some grid of the W values, we are able to trace aparametric plot with the abscissa PA(Js)/P, and the ordinate

δPA

PA= PA(Js + �±) − PA(Js)

PA(Js), (79)

where the upper branch will be given by �+ and the lower branchby �−. Similarly, the relative variations of the binary B period arecomputed according to

δPB

PB= PB(Js + �±) − PB(Js)

PB(Js). (80)

The dots in Figs 2 and 3 represent the values obtained by thisrecipe. Fig. 2 refers to a completely symmetric system with all fourmasses equal and the eccentricity of the principal orbit e = 0. Bysymmetry, the results for δPA and δPB are exactly the same, save forthe opposite signs, so only the plot for δPA/P is given. In the secondtest case, shown in Fig. 3, the masses differ. The ratios μ = 0.358,μA = 0.475, and μB = 0.193 refer to the KIC 4247791 system tobe studied in Section 5. Again, the principal orbit is circular. The

Figure 4. Top: Comparison of the numerical integration of the 1:1 resonantsystem (black line) with the isocontours of one-dimensional HamiltonianK from equation (47) (grey lines) in the phase space of resonant vari-ables: ϕ = λA − λB at the abscissa and J = �1/ at the ordinate (J∗� 0.0682836 from equation 57). The bold grey line is the separatrix de-limiting the 1:1 resonant zone about the stable equilibrium at ϕ = 90◦.Mean orbital elements, constructed by the procedure described in the text,are used for the numerically constructed trajectory. Bottom: Time depen-dence of the resonant angle ϕ = λA − λB indicating libration period ofabout 32.3 yr. The initial data of the numerical run used mean orbital pe-riods PA = 4.0747 d, PB = 4.0759 d, P = 400 d, zero eccentricities eA

and eB, and the eccentricity e = 0.25 for the relative orbit of the twobinaries. All stellar masses were set to 1.5 M� (completely symmetricsystem).

maximum variation of PB is bigger than that of PA, because thesystem has a smaller mass; the ratio of amplitudes is C−1 ≈ 4.22,as expected.

The numerically found values are confronted with simple es-timates (71) and (70), traced as the solid lines in Figs 2 and 3,respectively. The 5

3 power law holds well in both cases and thecoefficients are sufficiently accurate. Of course, there is some de-viation from the symmetry between positive and negative branchesin the numerically computed values for the second test case, absent



Figure 5. Semimajor axes aA and aB (left) and eccentricities eA and eB (right) for the two binaries A and B in 1:1 resonance (aA, ∗ = aB, ∗ � 0.072 013 65 au).The osculating values from numerical simulation of the full-fledged problem are shown in grey. The black lines are the mean elements, semimajor axes onthe left and eccentricities on the right, constructed using the analytical formulas for the quadrupole short-period effects given in the Appendix A. The meaneccentricities were constructed to be zero; their small values are due to limitations of the quadrupole corrections. The initial data of the numerical run usedmean orbital periods PA = 4.0747 d, PB = 4.0759 d, P = 400 d, zero eccentricities eA and eB, and the eccentricity e = 0.25 for the relative orbit of the twobinaries. All stellar masses were 1.5 M�.

in the approximate rules (71) and (70), but it occurs on the level ofat most third significant digit.

5 C OMPARISON W ITH FULL-FLEDGEDN U M E R I C A L I N T E G R AT I O N

In order to justify the above-outlined theory and demonstrate it isan adequate description of the resonant motion for specific systems,we now compare its results with direct numerical integration ofthe complete equations of motion of a 2+2 quadruple system. Thefull-fledged simulation uses a Hamiltonian in the original Jacobi co-ordinates (rA, rB, R) and the corresponding momenta ( pA, pB, P),in a form more general than (3) – with the potential not expandedin powers of the distance ratios. The system is propagated using theBulrisch–Stoer method with an adaptive time step to secure highaccuracy.

In the absence of the observed systems in an evident 1:1 res-onance, we borrow some parameters from the study of Lehmannet al. (2012). In particular, we consider a coplanar four-body systemin 2+2 architecture with orbital periods PA and PB near 4.0753 d,the mean of the true values observed in KIC 4247791. As expected,in order to be in the resonance, the modelled PA and PB need tobe closer to the representative value 4.0753 d for realistic valuesof the outer period P. To start with, we take PA = 4.0747 d andPB = 4.0759 d, instead of the observed values PA = 4.04973 d andPB = 4.10087 d. The outer orbit period P is not known in the KIC4247791 system, but it is possibly long. For sake of our illustration,we take P = 400 d. If it were longer (or even much longer), it wouldonly mean that the resonance width is smaller, implying PA andPB be closer to each other. The eccentricity of the outer orbit e is0.25, but merely the same results are obtained for all reasonablysmall e values (not approaching the stability limit of the quadruple

system). In order to illustrate different dynamical effects, we find itinteresting to first consider an equal-mass situation before we com-pare its results with the cases where the best-fitting stellar massesfrom Lehmann et al. (2012) are used. Specifically, we start with asituation where all four masses of the stars in A and B systems are1.5 M�.

Our aim is to consider binary orbits with zero mean eccentricitieseA and eB. However, we know the mutual interactions necessar-ily excite small eccentricity values. At this moment, it is usefulto recall the relation between the mean and osculating orbital ele-ments, briefly outlined in the Appendix A. Each of the orbital ele-ments e = (aA, kA, hA, λA; aB, kB, hB, λB; a, K,H, λ) is subject tothe short-period perturbations δe. The osculating values eo are thuscomposed of the mean values em and δe: eo = em + δe. What weaim to work with, and compare to the simple Hamiltonian theory ofthe resonance, are the mean elements em. For instance, the orbitalperiods PA, PB, P (and the related semimajor axes), or the eccen-tricity e mentioned above are the initial mean values. The sameway, our intended mean values of both eA and eB are zero. But topractically set the initial conditions of our numerical run, we needthe initial osculating values of the elements and from them the Ja-cobi coordinates and momenta. Therefore, we need to add δe toour specified values of periods/semimajor axes and eccentricities.These corrections have been worked out in the Appendix A. Tomake the situation simplest, we use the initial mean λA = 180◦,� A = 0◦, λB = 90◦, � B = 90◦, λ = 90◦, and � = 270◦. In thisconfiguration, the correction of secular angles (� A, � B, � ) arezero, and only eccentricities are given small non-zero values (e.g.eA = δkA from equation A6 and eB = δhB from equation A7). Ad-ditionally, the longitudes λA and λB are such that the system startswith the resonant angle ϕ = 90◦, corresponding to the stable equi-librium if J = Js, or to the maximum difference between J and its



Figure 6. The same as in Fig. 4, but now for a 2+2 system with the samemasses as in the KIC 4247791 system: m1 = 1.70 M�, m2 = 1.54 M�,m3 = 1.46 M�, and m4 = 1.35 M� (see Lehmann et al. 2012). Thereference level for the resonant momentum is now J∗ � 0.0767632 (toppanel).

equilibrium value Js on a given orbit, which is actually the case.The short-period corrections to the principal orbit elements are lessimportant, but also available in the Appendix (Section A2).

Setting the initial data, we numerically propagated the completesystem for time interval of 200 yr, outputting the Jacobi state vectors(rA, rB, R; pA, pB, P) every 6 hr. These were straightforwardlyused to compute osculating orbital elements eo. However, as weneed to compare the mean, rather than osculating, orbital elementsto the analytical theory from Sections 3 and 4, we subtracted theshort-period perturbations δe provided in the Appendix A to obtainthe mean orbital elements em. Finally, the mean resonant variables Jand ϕ are computed using the mean orbital elements. The momen-tum J was evaluated from aA and a, according to equation (44), andϕ as a difference λA − λB.

Figs 4 and 5 show the results. In the top part of Fig. 4, wecompare the phase-space trajectory of the numerically determinedresonant variables (ϕ, J) (black line) with the isocountours ofthe resonant Hamiltonian K from equation (47). The correspon-

dence is very good, implying the resonant Hamiltonian is anadequate model of the 1:1 resonance in this case. The small wigglesin the numerical solution reflect imperfection in our construction ofthe mean orbital elements, most likely because formulas for δe useonly the quadrupole short-period corrections. Both panels of Fig. 4show that the resonant angle ϕ librates about the equilibrium at 90◦

with an amplitude of nearly 45◦. The libration period is �32.3 yr,which favourably compares with the simple lower limit estimate �27.4 yr from equations (73) and (74). The slightly longer value isdue to finite, and non-negligible, amplitude of libration. Because theinitial conditions were at ϕ = 90◦, the difference of the initial meanperiods PA and PB from their average value PA = PB � 4.0753 dquantitatively characterizes the possible extent of the resonant state.We find (PA − PA)/PA � 1.47 × 10−4. Indeed, the resonance width� � 4.65 × 10−6 (see the top panel of Fig. 4), so that 3�/J∗ �2.03 × 10−4. Using equation (69), we note that the fractional differ-ence of the initial mean periods from their mean may be increasedby about 35 per cent to remain possibly locked in the 1:1 resonance,but not more. As a result, the maximum fractional difference in pe-riods PA and PB to characterize the resonant case is � 4 × 10−4, farsmaller than �1.2 × 10−2. Therefore, assuming the relative orbitof the binaries A and B has a period of a year or longer, the systemKIC 4247791 is near, but not in the 1:1 mean motion resonance.

Fig. 5 shows time dependence of the semimajor axes and ec-centricities of the binaries A and B. The grey symbols are theosculating elements, the black symbols are the mean elements re-sulting from subtraction of the short-period terms. As expected fromequation (A3), the short-period effects in semimajor axes aA andaB have the shortest periods �PA and �PB, but their amplitudeis the largest when the outer orbit is at pericentre (this is becauseof the ∝ (a/R)3 factor in equation A3). The short-period effectsin the eccentricities eA and eB have primarily the period P of theouter orbit; this is because of the constant term in the square-rootfactor in equation (A8). The mean values of the semimajor axesfairly well follow the short-period-averaged trend of the osculatingvalues and oscillate with the period of �32.3 yr. This is obviouslythe libration, resonant term. Its amplitude is about the same as thatof the short-period effects in this case. The mean eccentricities ofboth A and B orbits remain very small, less than �1.2 × 10−5.Their non-zero values reflect a slight inadequacy of our formulasfor the short-period perturbations, perhaps because contribution ofhigher-order than quadrupole terms near the pericentre of the outerorbit.

Figs 6 and 7 show the same as Figs 4 and 5, but now from anumerical run where we used non-equal masses of the four stars insystems A and B (otherwise, all initial data were the same). In partic-ular, we used the best-fitting values m1 = 1.70 M�, m2 = 1.54 M�,m3 = 1.46 M�, and m4 = 1.35 M� from Lehmann et al. (2012).As it is typical for the quadruple systems in 2+2 hierarchy, themasses are unequal, but not significantly different. Fig. 6 showsthat the resonant dynamics are not affected by such a small changein masses of participating stars. However, a difference is seen inthe behaviour of eccentricities eA and eB in the right-hand panelsof Fig. 7 (the semimajor axes aA and aB are again basically thesame as before). This change with respect to the idealized equal-mass case considered above is due to the activation of the octupoleterm in the secular coupling of the two binaries (n = 3 term inequation 23). The octupole term is well known to excite the ec-centricities (see the Appendix B and Vokrouhlicky 2016). Luckily,the effect is quite small in our case due to two effects: (i) the massparameters μA and μB are not much different from one half, suchthat the octupole strength factors 1 − 2μA and 1 − 2μB are small,



Figure 7. The same as in Fig. 5, but now for a 2+2 system with the same masses as in the KIC 4247791 system: m1 = 1.70 M�, m2 = 1.54 M�,m3 = 1.46 M�, and m4 = 1.35 M� (see Lehmann et al. 2012). The reference semimajor axes are now aA,� � 0.073 8842 au and aB,� � 0.070 4590 au.Non-equal masses in the binaries activate the secular octupole interaction. The principal consequence is the secular perturbation of the eccentricities eA andeB as described in the Appendix B. The black curves are again the mean eccentricities determined by our procedure of quadrupole short-period perturbationsubtraction from the osculating values; red is our simplified solution (equation B10).

and (ii) the system is not too compact, namely the fraction aA/a� aB/a � 0.037 is small (generally the factor by which multipolarterms decrease). This implies that the simple approximation devel-oped in the Appendix B provides fairly accurate result. Indeed, themean eccentricities of the A and B systems, constructed as usualby subtracting the short-period terms from the osculating values(black symbols in Fig. 7), are rather well matched by the prediction(equation B10) from our simple theory outlined in the Appendix B.The agreement will obviously become less satisfactory for morecompact quadruple systems, P/PA smaller, and/or cases with largerdifferences masses of participating stars, 1 − 2μν larger.

Our previous analysis showed that making the system more com-pact, namely assuming P smaller, would increase the resonant width.Thence, a question arises – possibly of a theoretical, rather than re-alistic nature – whether it could increase enough to match the orbitperiods observed in the KIC 4247791 system. For sake of a test, wethus re-ran our previous simulation with the following modification:(i) the orbital periods of binaries A and B were set PA = 4.100 871 dand PB = 4.049 732 d (those observed in the KIC 4247791 systemLehmann et al. 2012), and (ii) assumed the principal orbit with pe-riod P = 45 d and eccentricity e = 0.1. Note that this period is wellabove the limit required for the long-term stability of the system(e.g. Mardling & Aarseth 2001; Sterzik & Tokovinin 2002). Again,all stellar masses are as in the KIC 4247791 system. Results areshown in Figs 8 and 9. Libration of the resonant angle ϕ (Fig. 8)indicates that the system is locked in the resonance. In our example,the libration amplitude is quite large, but pushing P to still smallervalues could make the libration amplitude smaller. The librationperiod now decreased to �1.37 yr, as expected from a very compactquadruple system. Since the system remains in the 1:1 resonance,the semimajor axes aA and aB show the same pattern as before,but the eA and eB are different. Quite stronger coupling of the two

binaries now implies that the amplitude of the short-period effectsis �0.025. This is larger than the forced eccentricity due to sec-ular octupole coupling, which explains why the secular pattern isnot seen this time. Still, the eccentricities eA and eB remain smallenough.

We conclude that the observed orbital periods and stellar massesreported for the system KIC 4247791 are not incompatible withthe 1:1 mean motion resonant state as such (assuming coplanarity).However, the orbital period of their relative motion would need to beshort enough (≤45 d, say). It is to be seen if this requirement is notin conflict with other observational data of this interesting system.For instance, the very similar value of the systemic velocities ofthe A and B systems reported by Lehmann et al. (2012) imply verylikely a quite longer period P. It is also not clear whether the short Pvalue would not produce unobserved eclipse time variations in theKepler photometry. So our simulation demonstrates a possibility ofa system to be locked in a 1:1 resonance even with about a per cent-level difference in periods PA and PB, but it would actually applyonly if other observational facts corroborate this model (unlikely inthe case of KIC 4247791 system).

6 R E S O NA N C E C A P T U R E D I S C U S S I O N

A fundamental question concerning a resonance is the presence orabsence of some mechanism locking the system in such a particularstate. Without a capture mechanism, resonance may only appearrandomly, like a winning lottery ticket, or as a temporary phaseof evolution, like a particular number in a long countdown. Sincethe systems in a 1:1 resonance do not seem to be prolific amongstquadruple systems, let us inspect the problem according to theclassical scenario discussed by Henrard (1982).



Figure 8. The same as in Fig. 6, but now for a 2+2 system with the samemasses and orbital periods as in the KIC 4247791 system: m1 = 1.70 M�,m2 = 1.54 M�, m3 = 1.46 M�, m4 = 1.35 M�, PA = 4.100 871 d, andPB = 4.049 732 d (see Lehmann et al. 2012). The relative orbit has beengiven a period P = 45 d and eccentricity e = 0.1. The reference level for theresonant momentum is now J∗ � 0.159 0715 (top panel).

Similarly to Henrard, we are going to use a one degree of free-dom Hamiltonian with a slowly varying parameter, which makesthe system non-conservative, yet amenable to adiabatic approxima-tion. For the qualitative discussion, we can use the pendulum-likeHamiltonian (67) with two modifications: the momentum D is re-placed with an explicit form (J − J∗), and the approximate value(59) is substituted in all occurrences of J∗. The result is

K�(ϕ, J ; W ) = α

W 4

(J − W

1 + C

)2

+ βW 8 cos 2ϕ, (81)

where W-independent coefficients are

α = −3

2

(1 + C)5A

C, β = C4A4

(1 + C)8. (82)

Further, let us assume that W, is the slowly varying parameter ofthe Henrard’s model, with W = const. Since all other parameters

(in particular the mass ratios) are held fixed, the drift of W must berelated to secular change of the orbital period PB in equation (45).

At this point, we have to admit that technically the variability ofW as one of the canonical variables and not a physical parameter is tosome extent an abuse of the Henrard’s model, because it implies theaction of some force not having a potential function (so, impossibleto include in the Hamiltonian function). But the essence of heuristicreasoning shown below is more general and actually agrees withnumerical tests we performed to validate the conclusions (numericalintegration of equations 55 and 56 with a slowly varying W).

According to Henrard (1982), the fate of motion, once it getsclose to the separatrix region, is determined by the values of energyincrements accumulated during the motion in the close neighbour-hood of an upper separatrix (the values of momentum J∗ < J ≤J∗ + �) and its lower counterpart (J∗ − � ≤ J < J∗). Since bothvalues can be well approximated by the integrals along the separa-trices, we need a separatrix equation, giving J as a function of ϕ,W, and the Hamiltonian value E� = K�(0, W/(1 + C); W ) = βW 8.Solving a simple quadratic equation K�(ϕ, J ; W ) = E� for J, oneeasily finds

J± = W

1 + C±

√2β

αW 6| sin ϕ|. (83)

Then, the resulting energy increments are

B1 = −W

∫ 0

π

∂J+∂W

dϕ = W

(π

1 + C+ 12

√2β

αW 5

), (84)

for the upper separatrix motion, and

B2 = −W

∫ π

0

∂J−∂W

dϕ = W

(− π

1 + C+ 12

√2β

αW 5

), (85)

for the lower separatrix. Compared to the original Bi definition ofHenrard (1982), our integration limits differ in two aspects. First,having 2ϕ in the Hamiltonian (81), we integrate over the interval ofthe length π, which is a minor issue. More important is the swap ofthe upper and lower limits that adjusts to the opposite direction ofmotion along the upper and lower separatrices with respect to theHenrard’s model. The readers should note that Henrard consideredthe case, where the stable equilibrium is the local minimum of theHamiltonian function, whereas the stable equilibrium at ϕ = π/2 isthe maximum of our K�, which implies a different reasoning basedon the signs of B1 and B2.

Consider the decrease of PB as an expected outcome of tidalinteractions (or another dissipative mechanism) in the binary B.The sign of each Bi is typically determined by (−W ), because inthe range of W values we consider, the second term in the bracketsis smaller than the first one. Thus, B1 < 0 and B2 > 0. In thesecircumstances:

(i) Any trajectory originating above the upper separatrix withϕ = 0 will arrive at ϕ = π with the Hamiltonian value decreased by|B1| (i.e. with B1 < 0 added) and migrate away from the separatrix(in top panel of Fig. 4 – upwards). Capture does not occur.

(ii) A trajectory departing from a phase plane point below thelower separatrix, say J � W/(1 + C) and ϕ = π, will arrive atthe vicinity of ϕ = 0 with the Hamiltonian value increased by B2,hence, inside the libration zone. But then the motion below theupper separatrix decreases the Hamiltonian value by |B1|, whichmeans a drift outwards the libration zone. The net effect

B1 + B2 = 24 W

√2β

αW 5, (86)



Figure 9. The same as in Fig. 7, but now for a 2+2 system with the same masses and orbital periods as in the KIC 4247791 system: m1 = 1.70 M�,m2 = 1.54 M�, m3 = 1.46 M�, m4 = 1.35 M�, PA = 4.100 871 d, and PB = 4.049 732 d (see Lehmann et al. 2012). The reference semimajor axes are nowaA,� � 0.073 9404 au and aB,� � 0.070 4749 au. The relative orbit has been given a period P = 45 d and eccentricity e = 0.1. Compactness of the system nowmakes short-period effects stronger, exciting osculating eccentricities (grey symbols) up to few per cent. The mean eccentricities (black symbols) are about anorder of magnitude smaller, but non-zero.

inherits the sign of W , so the Hamiltonian value per one cycle isdecreased. Accordingly, after a short visit to the libration zone, thetrajectory continues the drift upwards, like in the case (i). Again,capture does not occur.

(iii) For the same reasons as in case (ii), a trajectory with theorigin at the outskirts of the libration zone (but inside it) will exitupwards.

Thus, when the dissipation decreases W, the capture into permanent1:1 resonance does not occur in our model.

The case of W > 0 may eventually lead to a capture with proba-bility (Henrard 1982)

Pc = B1 + B2

B1, (87)

which is fairly small. For a completely symmetric system withe = 0, and W < 0.3, it is below 0.02, and scales as W5. Moreover,it is difficult to identify a proper physical scenario resulting in theincrease of W.

In a similar manner, we have verified that the case of mass transferbetween the components of one of the binaries (this scenario strictlyadheres to the assumptions of the Henrard’s model) results in B1 andB2 having opposite signs, with a small sum B1 + B2, that leads tothe low probability of capture, proportional to W5. Thus, capture inthe resonance caused by the variation of μA or μB is also unlikely,although it might happen for some particular values of μA + μB.

7 C O N C L U S I O N S

In the present work, we dealt with a resonant configuration of aquadruple stellar system in 2+2 architecture. In particular, we con-

sidered the case where the mean orbital periods PA and PB are veryclose to each other hence constituting 1:1 mean motion resonance.As typical in orbital mechanics, the resonant problem has been re-duced to 1-degree Hamiltonian case allowing semi-analytical esti-mates of resonant width and several other interesting parameters. Acomparison with direct numerical integration of the complete prob-lem allowed us to justify the simplified model. We found that phasespace occupied by the 1:1 resonance in 2+2 quadruple shrinks tovery small volume for realistic ratio of the outer orbit period P andperiods PA or PB of the binaries. Additionally, adiabatic changesof PA or PB driving them towards the resonant configuration re-sult in only very small probability of capture. Therefore, number ofexisting 2+2 systems in the 1:1 resonance is expected to be verysmall.

In spite of the negative result, techniques and tools developed hereare suitable starting point for the next paper where we plan to dealwith the more complicated situations of the first-order resonances2:1 and 3:2 between PA and PB. As explained in the Introduction,there are at least two observed quadruple systems that were sug-gested to be locked in the 3:2 case. With the appropriate theoreticaltools available, we will be able to consider necessary parametricconditions for the resonance to exist and likelihood of the systemto be captured in this state.

AC K N OW L E D G E M E N T S

This work was partly funded through the research grant P209-13-01308S of the Czech Science Foundation. We thank the anonymousreviewer for valuable comments, especially for signalling the errorin equation (10) of Vokrouhlicky (2016), reproduced in the firstversion of the this paper.



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A P P E N D I X A : SH O RT PE R I O DP E RTU R BATI O N S

Linear, short period perturbations, albeit of low accuracy, can beuseful for the order of magnitude estimation, or help to establishinitial conditions for the numerical integration related to the spec-ified mean orbit of the secular system. The expressions below arederived from the generating function

S = S1 + S2, (A1)

where the term S1 is given by equation (33) and S2 by equa-tion (42). It is worth noting that S1 = ε3S2, so the short peri-odic terms resulting from the motion on the principal orbit shoulddominate.

Then, for any function of canonical variables, its osculating valesYo are, approximately, given by the Poisson bracket with SYo = Y + {Y ,S} = Y + δY , (A2)

where the expressions to the right are functions of the mean variables(solution of the system defined by K). Although using a simple sum(A1) we ignore the coupling term {S1,S2}, the loss is negligible (ofthe order ε10).

Once the Poisson bracket in (A2) has been evaluated (but notearlier !), we can set the mean eccentricities eA = eB = 0, accordingto the adopted model.

A1 Binaries a and b

Let ν refers to subscript A or B. Then the short periodic perturbationsin the semi-axes are

δaν = cν

3 n2a3 aν

2 n2νR

3cos 2(F − λν), (A3)

where cA = μ and cB = (1 − μ). For the mean longitudes,

δλν= − cν

n

nν

(f − l + e sin f

η3−21

8

na3

nνR3sin 2(F − λν)

), (A4)

the first term in the bracket, resulting from S2 should generallydominate, but for very small and/high eccentricity e, the secondterm may become important. This is because the first term strictlyspeaking vanishes for e = 0, and the second term may be stronglyamplified at the pericentre for large e value. It is of interest to notethat the first, long-period, term in equation (A4) has been extensivelyused for analysis of eclipse time variations in hierarchical triplesystems (see e.g. Soderhjelm 1975, 1982; Borkovits et al. 2003,2011, 2016; Rappaport et al. 2013). The second term has beenoverlooked and only recently briefly mentioned in Nemravova et al.(2016). Here, we give, for the first time, its formal derivation.

Due to the peculiarities of a perturbed small eccentricity orbit(e.g. Cohen & Lyddane 1981), the perturbations in eccentricitiesand longitudes of pericentres must be computed indirectly, throughnon-singular variables

kν = eν cos �ν, hν = eν sin �ν. (A5)

There we find

ko,ν = δkν = cν

4

n2a3

n2νR

3(9 cos (2F − λν)

− 2 cos λν + cos (2F − 3λν)) , (A6)

ho,ν = δhν = cν

4

n2a3

n2νR

3(9 sin (2F − λν)

− 2 sin λν sin (2F − 3λν)) , (A7)

and the osculating eccentricities

eo,ν =√

k2o,ν + h2

o,ν =

= cνn2a3

2√

2 n2νR

3

√43 − 20 cos 2(F − λν) + 9 cos 4(F − λν).

(A8)

The osculating longitudes of pericentres actually evolve quickly –the apsides rotate at the rate of nν , as seen from

tan �ν = ho,ν

ko,ν

. (A9)


http://dx.doi.org/10.1093/mnras/stv2530

http://dx.doi.org/10.1093/mnras/97.1.56



http://dx.doi.org/10.1093/mnras/97.5.388a

http://dx.doi.org/10.1007/BF01228961

http://dx.doi.org/10.1007/BF01235134

http://dx.doi.org/10.1093/mnras/stw784

http://dx.doi.org/10.1093/mnras/stv452

http://dx.doi.org/10.1007/BF01228839

http://dx.doi.org/10.1093/mnras/sts432

http://dx.doi.org/10.1046/j.1365-8711.2001.03974.x

http://dx.doi.org/10.1007/BF01844227

http://dx.doi.org/10.1093/mnras/stt1281

http://dx.doi.org/10.1088/0004-637X/768/1/33

http://dx.doi.org/10.1088/0004-637X/799/1/4

http://dx.doi.org/10.1111/j.1365-2966.2008.13613.x

http://dx.doi.org/10.1088/0004-6256/147/4/87

http://dx.doi.org/10.1093/mnras/stw1596

http://dx.doi.org/10.1093/mnras/sts616


Osculating mean anomalies merely oscillate around 0 or π, as foundfrom

lo,ν = λν − �ν + δλν. (A10)

A2 Principal orbit

Perturbations in the semi-axis of the principal orbit are

δa = da,A + da,B, (A11)

da,ν = μν (1 − μν)na2

νa2

2 nνR3

[nν

n

(1 −

(R

aη

)3)

− 3a2η

R2cos 2(F−λν)+9ea sin f

2Rηsin 2(F − λν)

](A12)

= −μν(1 − μν)3na2

ν

2nνacos 2(λ − λν) + O(e), (A13)

where equation (A13) can be used for sufficiently small eccentricityof the principal orbit. Short periodic perturbations in the meanlongitude are

δλ = dλ,A + dλ,B, (A14)

dλ,ν = μν(1 − μν)3a2

ν

4a2η4

[f − l +

(3 + η

4+ 1

1 + η

)e sin f

+ e2 sin 2f

2(1 + η)+ e3 sin 3f

12(1 + η)

− na3η3

nνR3

(3η

(1 − aη(R − aη2)

2R2(1 + η)

)sin 2(F − λν)

+ (R + aη2)e sin f

R(1 + η)cos 2(F − λν)

)](A15)

= −μν(1 − μν)9na2

ν

4nνa2sin 2(λ − λν) + O(e). (A16)

For the eccentricity and the longitude of pericentre, we find

δe = de,A + de,B, (A17)

de,ν = −μν (1 − μν)a2

ν

4a2e

[1

η− a3η2

R3

+ 3na4η

4nνR4

(4

(1 − R

a

)cos 2(F − λν)

1

η− e cos (F − 2λν + � ) + 5e cos (3F − 2λν − � )

)],

(A18)

and

δ� = d�,A + d�,B, (A19)

d�,ν = −μν (1 − μν)3a2

ν

4a2η4

[f − l +

(1

e+ 3e

4

)sin f

+ 1

2sin 2f + e

12sin 3f

− na3η3

nνR3

(3

4sin 2(F − λν) − 1

8sin 2(λν − � )

+ 5

8sin (4F − 2λν − 2� ) − 1

4esin (F − 2λν + � )

+ 7

4esin (3F − 2λν − � )

)](A20)

The above expressions for δe and δ� are valid only for sufficientlylarge eccentricity values (i.e. as long as the negative values of the os-culating eccentricity do not appear). For small mean eccentricities,non-singular variables

K = e cos �, H = e sin �, (A21)

should be used. Then, assuming the mean variables H = K = 0, weobtain

Ko = dK,A + dK,B, Ho = dH,A + dH,B, (A22)

with

dK,ν = μν(1 − μν)3a2

ν

4a2

[cos λ

− n

4nν

(7 cos (3λ − 2λν) + cos (λ − 2λν))

], (A23)

dH,ν = μν(1 − μν)3a2

ν

4a2

[sin λ

− n

4nν

(7 sin (3λ − 2λν) − sin (λ − 2λν))

]. (A24)

All the perturbation formulas can also be used as an analyticalfilter of the observed elements. The approximate values of the meanelements result from the subtraction

Y = Yo − δY , (A25)

where osculating values are substituted into the right-hand side.

A P P E N D I X B : SE C U L A R E VO L U T I O N O FI N N E R E C C E N T R I C I T I E S

Secular interaction of the two binary systems A and B may besolved analytically in the crudest approximation of nearly circularand coplanar orbits. While very restrictive, it is useful to understandsome of our results in Section 5.

Denote eA, eB, and e vectors pointing to the pericentre of theorbits A, B and mutual orbit with lengths equal to the appropri-ate eccentricities eA, eB, and e. Such normalized Laplace vectorsrepresent half of Milankovitch-type non-singular variables used byVokrouhlicky (2016) for description of the secular evolution ofthe 2+2 quadruple systems. Retaining only the linearly-averagedquadrupole (n = 2) and octupole (n = 3) interaction terms Hint inequation (23), we obtain (see Vokrouhlicky 2016, for details; ν = Aor B as before)

deν

dt= ων k ×

[eν − 2γν

η2e]

, (B1)

where k is the normal unit vector to the orbital plane of the quadru-ple system. We also denoted (as usual η = √

1 − e2, cA = μ, andcB = 1 − μ)

ων = 3

4

n2

nν

cν

η3, (B2)

and

γν = 5

8(1 − 2μν)

(aν

a

). (B3)

Assuming small inner eccentricities, the quadratic terms in eA

and eB have been neglected in the right-hand sides of equa-tions (B1). For exactly equal-mass binaries A and B, thence1 − 2μA = 1 − 2μB = 0, these equations are autonomous andprovide a trivial time evolution of eA and eB: a constant circulation



in the plane normal to k with frequencies ωA and ωB. A small mis-match in masses of the binaries activates the role of the octupoleinteraction and makes time evolution of eA and eB coupled to thetime evolution of their relative-orbit pericentre vector e. Neglect-ing terms of the order of eccentricities eA and eB in the right handside of de/dt , we conclude that e steadily circulates about k withfrequencies ωAB, whose quadrupole approximation provides

ωAB � 3

4

n

a2η4

[μA (1 − μA) a2

A + μB (1 − μB) a2B

]. (B4)

In order to proceed with secular eccentricity solution of the innersystems A and B, we set k = (0, 0, 1) and e = (K,H, 0), whereK = e cos � and H = e sin � . As mentioned above e is constantand � = ωABt + � 0. Denoting eν = (kν, hν, 0), we now writeequation (B1) in the following form:

dkν

dt= −ων

[hν − 2

eγν

η2sin �

], (B5)

dhν

dt= ων

[kν − 2

eγν

η2cos �

]. (B6)

This is a simple system describing two-dimensional forced har-monic oscillator with proper frequency ων . Assuming, in partic-

ular, initial conditions at origin, notably eν = (0, 0, 0) for t = 0,we obtain

kν(t) = −e�,ν [cos (ωνt + �0) − cos � ] , (B7)

hν(t) = −e�,ν [sin (ωνt + �0) − sin � ] , (B8)

where

e�,ν = 2eγν

η2

1

1 − ωABων

. (B9)

This is the forced eccentricity by the presence of the octupole termin the interaction of systems A and B, replacing the zero-eccentricitystationary state in the quadrupole model.

From equations (B7) and (B8), we easily obtain secular evolutionof the system A eccentricity

eν(t) =√

2e�,ν

√1 − cos (ων − ωAB) t . (B10)

The eccentricity reaches a maximum 2e�,ν . A consistency of thesolution obviously requires that this quantity remains small.

This paper has been typeset from a TEX/LATEX file prepared by the author.


resonant dynamics of gravitationally bound pair of binaries...

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